Take away the start

The idea

The sum of the consecutive natural numbers from m + 1 to n is Sₙ – Sₘ, the sum of the first n natural numbers minus the sum of the first m.

A sum that does not start at 1

The book asks for 25 + 26 + 27 + … + 58. That is 34 numbers to add, and the run does not begin at 1.

Which trick could help?

• Find 1 + 2 + … + 58, then take away 1 + 2 + … + 24
• Add only the first and the last number
• Multiply 25 by 58

Cut away the start

The bars are 1, 2, 3, … up to the last number. The grey bars are taken away. Type the first and the last number of a run.

Sₙ − Sₘ

The sum of the consecutive natural numbers from m + 1 to n is Sₙ − Sₘ: the sum of the first n natural numbers minus the sum of the first m.

Book example: 25 + 26 + … + 58 = (1 + 2 + … + 58) − (1 + 2 + … + 24) = S₅₈ − S₂₄

S₅₈ = 58 × 59 ÷ 2 = 1711 and S₂₄ = 24 × 25 ÷ 2 = 300. So the sum is 1711 − 300 = 1411.

Notes

The sum of the consecutive natural numbers from m + 1 to n is Sₙ − Sₘ, the sum of the first n natural numbers minus the sum of the first m.

Check yourself

Find 5 + 6 + 7 + 8 + 9 + 10. (S₁₀ = 55 and S₄ = 10.)

Answer: 45

S₁₀ − S₄ = 55 − 10 = 45. Check: 5 + 6 + 7 + 8 + 9 + 10 = 45.

To find 25 + 26 + … + 58 which subtraction do we use?

Find 31 + 32 + … + 40. (S₄₀ = 820 and S₃₀ = 465.)

Answer: 355

S₄₀ − S₃₀ = 820 − 465 = 355.

S₁₀ = 55 and S₂₀ = 210. What is 11 + 12 + … + 20?

Answer: 155

S₂₀ − S₁₀ = 210 − 55 = 155.

• S₅₈ − S₂₅. S₂₅ also contains 25, but 25 belongs to the run. Remove only the first 24 numbers.
• S₅₈ − S₂₄ — correct. Yes. The run starts at 25, so we remove the sum of the first 24 numbers.
• S₅₈ − S₃₃. 58 − 25 = 33 counts the steps, not the sum. We must subtract the sum S₂₄.